Let $G$ be a compact group and let $\widehat{G}$ denote the set of all equivalence classes of irreducible representations of $G$. For each $\pi \in \widehat{G}$, we use $\chi_\pi$ to denote the character of $G$ constructed by $\pi$.
Let $x$ be an element in $G$ so that $|d_\pi^{-1}\chi_\pi(x)|=1$ for every $\pi \in \widehat{G}$ and $$ \frac{d_\rho^{-1} \chi_\rho(x)}{d_\pi^{-1} \chi_\pi(x)\ d_\sigma^{-1} \chi_\sigma(x)} >0 $$ when $\rho$ appears in the tensor decomposition of $\pi \otimes \sigma$ for every $\pi, \sigma, \rho \in \widehat{G}$. Does it imply that $x$ is the identity of the group $G$?
The first condition implies that $x$ is central (since it implies, by looking at the eigenvalues of $x$ acting on irreps, that $x$ acts by a scalar in every irrep), and if $x$ is central then the second condition always holds: $x$ acts by some scalar on $\pi$ and some other scalar on $\sigma$, and so it acts by the product of those scalars on every irrep in $\pi \otimes \sigma$. So if $x$ is central then that quotient is always $1$.